Graph manifolds are compact orientable 3–manifolds obtained by gluing several copies of D2×S1 and N2×S1 together by homeomorphisms of some components of their boundaries (D2 is the 2–disc and N2 ...denotes the 2–disc with two holes). Here we study spines and surgery representations of orientable graph manifolds, and derive geometric presentations of their fundamental group. Then we determine the homeomorphism type of many Takahashi manifolds and the Teragaito manifolds, showing that they are graph manifolds with specified invariants. Finally, we describe graph manifolds arising from toroidal surgeries on certain classes of hyperbolic knots.
The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed manifolds up to piecewise-linear (PL) homeomorphism. For this, we use ...the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the reduced complexity, for any closed $n$-dimensional PL manifold. Here we consider this invariant, and extend in this context the concept of average order first introduced by Luo and Stong in 1993, and successively investigated by Tamura in 1996 and 1998. Then we obtain some classification results for closed connected smooth low-dimensional manifolds according to reduced complexity and average order. Finally, we answer to a question posed by Trout in 2013.
We study closed connected orientable 3-manifolds obtained by Dehn surgery along the oriented components of a link, introduced and considered by Motegi and Song (2005) and Ichihara et al. (2008). For ...such manifolds, we find a finite balanced group presentation of the fundamental group and describe exceptional surgeries. This allows us to construct an infinite family of tunnel number one strongly invertible hyperbolic knots with three parameters, which admit toroidal surgeries and Seifert fibered surgeries. Among the obtained results, we mention that for every integer
n
>
5
there are infinitely many hyperbolic knots in the 3–sphere, whose
(
n
-
2
)
and
(
n
+
1
)
-surgeries are toroidal, and
(
n
-
1
)
and
n
-surgeries are Seifert fibered.
In this paper we continue our investigations of 4-dimensional complexes in A. Cavicchioli, F. Hegenbarth, F. Spaggiari,
, Mediterr. J. Math.
(2020), 175. We study a class of finite oriented ...4-complexes which we call
-complexes, defined as follows. An
is a 4-dimensional finite oriented
-complex
with a single 4-cell such that
with a fundamental class
. By well-known results of Wall, any Poincaré complex is of this type. We are interested in two questions. First, for which 3-complexes
does an element
exist such that
is a Poincaré complex? Second, if there exists one, how many others can be constructed from
? The latter question was addressed studied in the above cited previous paper of the authors. In the present paper we deal with the first problem, and give necessary and sufficient conditions on
and
to satisfy Poincaré duality with
- and Λ-coefficients. Here Λ denotes the integral group ring of
. Before, we give a classification of all
-complexes based on the finite 3-complex
, and make some remarks concerning
- and Λ-Poincaré duality.
Abstract
In this paper we continue our investigations of 4-dimensional complexes in A. Cavicchioli, F. Hegenbarth, F. Spaggiari,
Four-dimensional complexes with fundamental class
, Mediterr. J. Math.
...17
(2020), 175. We study a class of finite oriented 4-complexes which we call
FC
4
{\mathrm{FC}_{4}}
-complexes, defined as follows. An
FC
4
{\mathrm{FC}_{4}}
-
complex
is a 4-dimensional finite oriented
CW
-complex
X
with a single 4-cell such that
H
4
(
X
,
ℤ
)
≅
ℤ
{H_{4}(X,\mathbb{Z})\cong\mathbb{Z}}
with a fundamental class
X
∈
H
4
(
X
,
ℤ
)
{X\in H_{4}(X,\mathbb{Z})}
. By well-known results of Wall, any Poincaré complex is of this type. We are interested in two questions. First, for which 3-complexes
K
does an element
φ
∈
π
3
(
K
)
{\varphi\in\pi_{3}(K)}
exist such that
K
∪
φ
D
4
{K\cup_{\varphi}D^{4}}
is a Poincaré complex? Second, if there exists one, how many others can be constructed from
K
? The latter question was addressed studied in the above cited previous paper of the authors. In the present paper we deal with the first problem, and give necessary and sufficient conditions on
K
and
φ
∈
π
3
(
K
)
{\varphi\in\pi_{3}(K)}
to satisfy Poincaré duality with
ℤ
{\mathbb{Z}}
- and Λ-coefficients. Here Λ denotes the integral group ring of
π
1
(
K
)
{\pi_{1}(K)}
. Before, we give a classification of all
FC
4
{\mathrm{FC}_{4}}
-complexes based on the finite 3-complex
K
, and make some remarks concerning
ℤ
{\mathbb{Z}}
- and Λ-Poincaré duality.
The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 5-manifolds up to piecewise-linear (PL) homeomorphism. For this, we use ...the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called
crystallizations
. Within this representation theory, Bracho and Montejano introduced in 1987 a nonnegative numerical invariant, called the
reduced complexity
, for any closed
n
-dimensional PL manifold. Here we obtain the complete classification of all closed connected smooth 5-manifolds of reduced complexity less than or equal to 20. In particular, this gives a combinatorial characterization of
S
2
×
S
3
among closed connected spin PL 5-manifolds.
We consider some families of two-bridge links, including the links
b
(
2
p
,
3
)
, the twisted Whitehead links and the double twist links, and calculate their character varieties. Then we give simple ...geometrical descriptions of such varieties, and determine the number of their irreducible components. Our paper relates to the nice work of Hilden, Lozano and Montesinos on the character variety of a class of 2-bridge links, but uses very different methods based on the concept of palindrome presentations of link groups. The obtained formulas for the above character varieties are new, easily programmable, and more simple in many cases than the equations known in the literature.
Four-Dimensional Complexes with Fundamental Class Cavicchioli, Alberto; Hegenbarth, Friedrich; Spaggiari, Fulvia
Mediterranean journal of mathematics,
12/2020, Letnik:
17, Številka:
6
Journal Article
Recenzirano
This paper continues the study of 4-dimensional complexes from our previous work Cavicchioli et al. (Homol Homotopy Appl 18(2):267–281, 2016; Mediterr J Math 15(2):61, 2018.
...https://doi.org/10.1007/s00009-018-1102-3
) on the computation of Poincaré duality cobordism groups, and Cavicchioli et al. (Turk J Math 38:535–557, 2014) on the homotopy classification of strongly minimal
PD
4
-complexes. More precisely, we introduce a new class of oriented four-dimensional complexes which have a “fundamental class”, but do not satisfy Poincaré duality in all dimensions. Such complexes with partial Poincaré duality properties, which we call
SFC
4
-complexes, are very interesting to study and can be classified, up to homotopy type. For this, we introduce the concept of resolution, which allows us to state a condition for a
SFC
4
-complex to be a
PD
4
-complex. Finally, we obtain a partial classification of
SFC
4
-complexes. A future goal will be a classification in terms of algebraic
SFC
4
-complexes similar to the very satisfactory classification result of
PD
4
-complexes obtained by Baues and Bleile (Algebraic Geom. Topol. 8:2355–2389, 2008).
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